# 2.7: Square Roots and Real Numbers

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Find square roots.
- Approximate square roots.
- Identify irrational numbers.
- Classify real numbers.
- Graph and order real numbers.

## Vocabulary

Terms introduced in this lesson:

- square root
- principal square root
- positive square root
- radicals
- perfect squares
- prime factors
- rational number
- irrational number
- approximate answer
- approximation
- non-repeating decimals
- simplest form
- integer
- sub-intervals

## Teaching Strategies and Tips

Contrary to what students think, there is no discrepancy between there being two possible values for \begin{align*}b\end{align*}, given a positive \begin{align*}x\end{align*}, such that \begin{align*}b^2 = x\end{align*} and only one value for \begin{align*}\sqrt{x}\end{align*}. The positive number \begin{align*}b\end{align*} is the *principal* square root. It is the value of the function.

Point out that \begin{align*}\surd^{\overline{\;\;\;}}\end{align*} and \begin{align*}\sqrt[2]{}\end{align*} mean the same thing. The \begin{align*}“2”\end{align*} is understood.

In Example 1:

- Use factor trees to break down the radicands into as many perfect squares as possible.
- Primes which appear an even number of times constitute a perfect square.
- Any unpaired factors are left under the radical sign.
- The convention is to leave any irreducible radical in the form: \begin{align*}\mathrm{(square\ root)} \sqrt{(\mathrm{irreducible \ part})}\end{align*}

Lots of practice early on makes it easier for students when the radicals become more complicated and involve variable expressions.

Use Example 3 to multiply, divide, and simplify radical expressions. State which rules were used in classroom examples.

In Example 4, students use a calculator and round their answer to three decimal places. Review rounding decimals.

Motivate irrational numbers in Example 5.

- Irrational numbers complete the set of real numbers.
- They cannot be expressed as ratio of two integers.
- They have an unending (non-terminating), seemingly random (non-repeating) decimal expansion.
- Some irrational numbers: \begin{align*}\pi, \frac{1} {\pi}, \sqrt{2}, \sqrt{3}, \sqrt{\mathrm{any \ prime}}\end{align*}

In Example 5, students identify the given numbers. A number is rational if it

- can be expressed as a fraction of integers.
- has a finite decimal expansion.
- has a repeating block of digits in its decimal expansion.

Have students check their calculator displays for *repeating blocks of digits,* not just patterns. For example, the two numbers below have obvious patterns, but no repeating blocks, and therefore are irrational:

- \begin{align*}0.010010001000010000010000001 \ldots \end{align*}
- \begin{align*}0.12345678910111213141516171819202122 \ldots\end{align*}

Remind students that integers can be written as fractions with a \begin{align*}1\end{align*} in the denominator. See Example 6a and 6b. The strategy in Example 6e is to simplify first.

The strategy in Example 8 is to use a calculator to find the decimal expansions of the numbers rounded to as many places as is needed to identify each. Since all the numbers are between \begin{align*}3.1\end{align*} and \begin{align*}3.2\end{align*}, going out to three decimal places is sufficient.

## Error Troubleshooting

In Problem 3 of the *Review Questions*:

- Students should find repeating blocks of digits before claiming that a number is rational.
- It is not sufficient to claim that the “unpredictable” decimals of a number on a calculator display belong to an irrational number. The decimal expansion of \begin{align*}2/19\end{align*} has \begin{align*}18\end{align*} seemingly random digits in its repeating block; therefore, the rational number \begin{align*}2/19\end{align*} would go “unchecked” on an ordinary calculator display because it could not show enough digits.